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Numerical modeling of resonant internal seiche triggered by unsteady withdrawal Gu, Li

Abstract

Most lakes and reservoirs are temperature stratified for some period during the year, at which time a typical fresh water body can be divided into a well-mixed, warm, surface layer called the epilimnion, which is separated from the colder and relatively stagnant bottom water, called the hypolimnion, by an intermediate region with strong temperature gradient known as the metalimnion, which in conditions of temperature stratification only, is sometimes called the thermocline. Internal standing waves or internal temperature seiches within a stratified lake or reservoir play an important role in the physical and biological dynamics. Most previous studies have considered wind-generated internal seiches. Little attention has been given to the problem of triggering internal seiche by unsteady withdrawal, which was first proposed by Imberger (1980) as a possible pump-back scheme to optimize the water quality in a lake or reservoir. The resonant internal seiche triggered by unsteady withdrawal is studied numerically in this thesis using the C/C (Cloud-In-Cell) method. The investigation is concerned with internal seiches in a viscous, incompressible and weakly stratified fluid which is confined within a rectangular tank with a horizontal line sink in an end wall. Two-layered fluid with a sharp interface and two-layered fluid with a diffuse interface are investigated. Two-layered fluid with a sharp interface approximates conditions of lakes and reservoirs in late summer or early autumn when the thermocline is quite thin, while two-layered fluid with a diffuse interface approximates conditions of lakes and reservoirs in spring and late autumn, when the thermocline can be quite thick. The CIC method used in present study combines the best features of both Lagrangian and Eulerian approaches, and is most advantageously applied to fluid dynamics problems with density stratification. The numerical model solves, in finite difference form, the governing equations in terms of stream function and vorticity, and therefore avoids the complexity of solving the pressure field. The free surface boundary conditions in this case are somewhat difficult to apply. However, for the case of small density variation, simplified free surface boundary conditions may be chosen without affecting the main features of the flow. A two-layered fluid with a sharp interface supports only one vertical internal wave mode. With the fluids in the upper and lower layers moving in the opposite directions, large velocity shear develops at the density interface. The introduction of a continuously stratified interfacial region between two homogeneous fluids enables more than one vertical internal wave mode to exist, although the energy is usually partitioned into the first few internal wave modes. It is found that the natural frequency of internal waves decreases with increasing interface thickness and viscosity. The natural periods of a two-layered fluid system with an exponentially stratified interfacial layer are well predicted by the numerical model, in comparison with the theoretical solutions of Fieldstad (1933). By discharging fluid from the tank, two distinct classes of internal waves are created, namely, forced and free internal waves. The forced internal wave follows the forcing discharge and sustains as long as the discharge is retained, while the free internal wave oscillates according to its natural frequency with decaying amplitude due to viscous damping. Periodic beating is observed as a result of the interaction between the forced and free internal waves. The resonant internal seiche is created when the frequency of unsteady forcing discharge corresponds to the natural frequency of internal waves.

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